Network science has been a rapidly evolving field to study systems made of interactions between entities. Studying the structure of such networks reveals indeed the underlying mechanisms of these systems, and has been proven successful in many domains, such as sociology, biology, or geography. Recently, connections between network science and signal processing have emerged, making the use of a wide variety of tools possible to study networks. In this chapter, a focus is made on a methodology introduced to transform a graph into a collection of signals, using a multidimensional scaling technique: by projecting a distance matrix representing relations between vertices of the graph as points in a Euclidean space, it is possible to interpret coordinates of vertices in this space as signals, and take advantage of this dual representation to develop new tools for the study of networks. Deeper considerations of this methodology are proposed, by strengthening the connections between the obtained signals and the common graph structures. A robust inverse transformation method is next described, taking into account possible changes in the signals. Establishing a robust duality between graphs and signals opens up new perspectives, as classical signal processing tools, such as spectral analysis or filtering, are made available for the study of the structure of networks.

We address two questions related to the notion of frequency and its possible extensions in the case of evolutive situations, some of them leading to paradoxes. We first make a distinction between the concepts of instantaneity and locality; we then discuss links between local and global spectral properties, in relation, in particular, with the recently introduced phenomena of “supershift” and “superoscillation.”

Joseph Fourier’s methods (and their variants) are omnipresent in audio signal processing. However, it turns out that the underlying ideas took some time to penetrate the field of sound analysis and that different paths were first followed in the period immediately following Fourier’s pioneering work, with or without reference to him. This illustrates the interplay between mathematics and physics as well as the key role played by instrumentation, with notable inventions by outsiders to academia, such as Rudolph Koenig and Édouard-Léon Scott de Martinville.

En partant de l'exemple tres simple du sinus cardinal vu comme fonction oscillante, on s'interesse a deux questions relatives a la notion de frequence. Dans un premier temps, on distingue les concepts d'instantaneite et de localite dans les cas evolutifs ; dans un second temps, on remet en perspective les proprietes spectrales locale et globale, en lien en particulier avec les phenomenes de "supershift" et "superoscillation".

: Developed by Joseph Fourier at the beginning of the 19th century in order to solve the equations that rule heat propagation, Fourier analysis found many applications which, in turn, motivated developments of this theory in several directions. At the end of the 1940s, the first computers made possible the development of signal and image processing. In his pioneering work, Dennis Gabor advocated for a localized Fourier transform, e.g. in sound processing ; he thus opened a vast research program which culminated in the 1980s with the discovery of orthonormal bases of this type (Malvar and Wilson bases) ; these bases, together with the corresponding fast algorithms, played a key role in the detection of gravitational waves ; additionally, the audio compression formats used in cell phones, are based on these decompositions. In the mid 1980s, ano-ther extension of Fourier analysis developed under the seminal impulsion of the geophysicist Jean Morlet, who was aware that local Fourier analysis performed poorly when analyzing seismic data. Together with the theoretical physicist Alex Grossman, he laid the foundations of wavelet analysis, where the signal is decomposed on the translations and dilations of a single function. The mathematical theory and the computational aspects of this theory followed from the joint efforts of Yves Meyer, St´ephane Mallat and Ingrid Daubechies, and led to other technological breakthroughs ; e.g. the format for high resolution professional photography compression, JPEG 2000, is based on such decompositions. We will expose the evolution of the mathematical ideas that started with the seminal impetus of Joseph Fourier, and are at the heart of today’s technological revolution.

Phenomena that exhibit long-range dependence characteristics have been found to exist in many different areas, such as physics, geology, communications and biology. This chapter shows that the Fano factor can be understood through multiresolution analysis and therefore to propose a timescale-based generalization of this tool. It explains how and why this wavelet analysis of long-range dependent point processes provides with a more versatile and powerful estimator of the long-range dependence parameter. The chapter presents the analysis of such a point process made of a spiketrain of discharges recorded from auditory neurons responding to an acoustic stimulus. The exponents of the power-laws, whatever the statistics are the most meaningful parameters characterizing the long-range dependence phenomenon. The Poisson process plays, for point processes, the reference role the white Gaussian noise does for continuous time random processes. The chapter aims to point out some connections between the standard and the wavelet-based Fano factor.

In this paper, we introduce the ASTRES* toolbox which offers a set of Matlab functions for non-stationary multi-component signal processing. The main purposes of this proposal is to offer efficient tools for analysis, synthesis and transformation of any signal made of physically meaningful components (e.g. sinusoid, trend or noise). The proposed techniques contain some recent and new contributions, which are now unified and theoretically strengthened. They can provide efficient time-frequency or time-scale representations and they allow elementary components extraction. Usage and description of each method are then detailed and numerically illustrated.

Si Monsieur Jourdain faisait de la prose sans le savoir, la plupart des scientifiques ont longtemps fait du traitement du signal sans en avoir une claire conscience.

The description of large temporal graphs requires effective methods giving an appropriate mesoscopic partition. Many approaches exist today to detect communities in static graphs. However, many networks are intrinsically dynamical, and need a dynamic mesoscale description, as interpreting them as static networks would cause loss of important information. For example, dynamic processes such as the emergence of new scientific disciplines, their fusion, split or death need a mesoscopic description of the evolving network of scientific articles. There are two straightforward approaches to describe an evolving network using methods developed for static networks. The first finds the community structure of the aggregated network; however, this approach discards most temporal information, and may lead to inappropriate descriptions, as very different dynamic data can give rise to the identical static graphs. The opposite approach closely follows the evolutions and builds networks for successive time slices by selecting the relevant nodes and edges, the mesoscopic structure of each of these slices is found independently and the structures are connected to obtain a temporal description. By using an optimal structural description at each time slice, this method avoids the inertia of the aggregated approach. The inherent fuzziness of the communities leads to noise and artifacts. Here, we present an approach that distinguishes real trends and noise in the mesoscopic description of data using the continuity of social evolutions. To be follow the dynamics, we compute partitions for each time slice, but to avoid transients generated by noise, we modify the community description at time t using the structures found at times t-1 and t+1. We show the relevance of our method on the analysis of a scientific network showing the birth of a new subfield, wavelet analysis.